Exotic R4

In mathematics, an exotic is a differentiable manifold that is homeomorphic (i.e. shape preserving) but not diffeomorphic (i.e. non smooth) to the Euclidean space The first examples were found in 1982 by Michael Freedman and others, by using the contrast between Freedman's theorems about topological 4-manifolds, and Simon Donaldson's theorems about smooth 4-manifolds.[1][2] There is a continuum of non-diffeomorphic differentiable structures of as was shown first by Clifford Taubes.[3]

Prior to this construction, non-diffeomorphic smooth structures on spheres – exotic spheres – were already known to exist, although the question of the existence of such structures for the particular case of the 4-sphere remained open (and still remains open as of 2023). For any positive integer n other than 4, there are no exotic smooth structures on in other words, if n ≠ 4 then any smooth manifold homeomorphic to is diffeomorphic to [4]

  1. ^ Kirby (1989), p. 95
  2. ^ Freedman and Quinn (1990), p. 122
  3. ^ Taubes (1987), Theorem 1.1
  4. ^ Stallings (1962), in particular Corollary 5.2

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